Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0905.4029 · Physical Review E

Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat

L. A. Fernández, A. Gordillo-Guerrero, Víctor Martín-Mayor, Juan Jesus Ruiz-Lorenzo

Published Nov 6, 2009• 8 citations
arXivHTMLPDF

Abstract

A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization (the so-called quotients method) to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides accurate determinations of the anomalous dimension, eta, and of the (Fisher-renormalized) thermal nu exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.

DOI
10.1103/physreve.80.051105

Related papers

Finite-size scaling in the canonical ensemble

Youjin Deng, Henk W. J. Blöte

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent $y_1$ that depends on the temperature exponent $y_t$ and the dimensionality $d$ as $y_1=-|2y_t-d|$. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.

arXiv:cond-mat/0508348•Aug 15, 2005•1 citation

Finite-size behaviour of the microcanonical specific heat

H Behringer, M Pleimling, A Hüller

For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.

arXiv:cond-mat/0411153•Journal of Physics A Mathematical and General•Jan 20, 2005•24 citations

Microcanonical renormalization-group simulation of Ising systems

Adauto J. F. de Souza, F. G. Brady Moreira

We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain ${\mathit{u}}_{\mathit{c}}$=-0.991(1), ${\mathit{T}}_{\mathit{c}}$=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).

Physical review. B, Condensed matter•Oct 1, 1993•11 citations

Phase Transitions in Disordered Systems

A. Gordillo-Guerrero

We face the problem of phase transitions in diluted systems both from theoretical and numerical sides. We study the effects of quenched site-dilution in classical models (Heisenberg, Ising and Potts) in 2, 3, and 4 dimensions both by using the Renormalization Group and numerical simulations in the canonical and microcanonical ensembles. We propose and check a new formulation of the Finite Size Scaling ansatz (FSS) inside the microcanonical ensemble. We use microcanonical simulations to obtain a clear fist-order behavior for the diluted Potts model in 3D, estimating the tricritical dilution. We perform large-scale simulations of the 3D diluted Heisenberg model, checking its self-averaging properties. Finally we study the 4D diluted Ising model obtaining from the FSS of the specific heat a clear differentiation between the existing conflicting theories. We also compiled a large number of appendix that we expect to be used as future reference.

arXiv:1004.1579•Mar 11, 2010•0 citations

Improved Phenomenological Renormalization Schemes

M. A. Yurishchev

An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1

•Jan 1, 2001•3 citations

Critical-point finite-size scaling in the microcanonical ensemble

Ann Bruce, Nigel B. Wilding

We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a significant role in determining the observed differences between the canonical and microcanonical specific heats of systems of finite size, in the critical region.

arXiv:cond-mat/9908025•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1999•14 citations
A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization (the so-called quotients method) to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides accurate determinations of the anomalous dimension, eta, and of the (Fisher-renormalized) thermal nu exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.
We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent y1y_1y1​ that depends on the temperature exponent yty_tyt​ and the dimensionality ddd as y1=−∣2yt−d∣y_1=-|2y_t-d|y1​=−∣2yt​−d∣. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.
For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.
We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain uc{\mathit{u}}_{\mathit{c}}uc​=-0.991(1), Tc{\mathit{T}}_{\mathit{c}}Tc​=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).
We face the problem of phase transitions in diluted systems both from theoretical and numerical sides. We study the effects of quenched site-dilution in classical models (Heisenberg, Ising and Potts) in 2, 3, and 4 dimensions both by using the Renormalization Group and numerical simulations in the canonical and microcanonical ensembles. We propose and check a new formulation of the Finite Size Scaling ansatz (FSS) inside the microcanonical ensemble. We use microcanonical simulations to obtain a clear fist-order behavior for the diluted Potts model in 3D, estimating the tricritical dilution. We perform large-scale simulations of the 3D diluted Heisenberg model, checking its self-averaging properties. Finally we study the 4D diluted Ising model obtaining from the FSS of the specific heat a clear differentiation between the existing conflicting theories. We also compiled a large number of appendix that we expect to be used as future reference.
An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1
We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a significant role in determining the observed differences between the canonical and microcanonical specific heats of systems of finite size, in the critical region.